On negation: Pure local rules

This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multiple-conclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual sub-classical rules involving this logical constant—among others, well-known forms of proof by cases, consequentia mirabilis and reductio ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness can thus be formulated, allowing for pseudo-scotus and ex contradictione to be differentiated and for a comprehensive version of the Principle of Non-Triviality to be presented. A final proposal is made to the effect that—pure positive rules involving negation being often fallible—a characterization of what most negations in the literature have in common should rather involve, in fact, a reduced set of negative rules.